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Question-196525




Question Number 196525 by Mastermind last updated on 26/Aug/23
Answered by mr W last updated on 27/Aug/23
(√(m^2 +1))>(√m^2 )=m  (√(m^2 +1))<(√(m^2 +2m+1))=(√((m+1)^2 ))=m+1  since m<(√(m^2 +1))<m+1,  ⇒[(√(m^2 +1))]=m
$$\sqrt{{m}^{\mathrm{2}} +\mathrm{1}}>\sqrt{{m}^{\mathrm{2}} }={m} \\ $$$$\sqrt{{m}^{\mathrm{2}} +\mathrm{1}}<\sqrt{{m}^{\mathrm{2}} +\mathrm{2}{m}+\mathrm{1}}=\sqrt{\left({m}+\mathrm{1}\right)^{\mathrm{2}} }={m}+\mathrm{1} \\ $$$${since}\:{m}<\sqrt{{m}^{\mathrm{2}} +\mathrm{1}}<{m}+\mathrm{1}, \\ $$$$\Rightarrow\left[\sqrt{{m}^{\mathrm{2}} +\mathrm{1}}\right]={m} \\ $$
Commented by Mastermind last updated on 27/Aug/23
Thank you so much
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much} \\ $$

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